Mechanical roller animation
How about using gravity? If you put gravity as z=10 (i.e. pushing upwards) and y=10 (pushing against back of stacker), then that would do the stacking. If you then animate the Physics/RigidBody/Settings/Dynamic toggle, you can turn gravity off for each object until the stacker reaches it.

You might want to use an invisible plane or two as passive rigid bodies to control the movement, as the stacker geometry is relatively complicated so might not work well as a collision object.